Page 463 - Aircraft Stuctures for Engineering Student

444 Structural constraint

some of the results of their analysis. We shall limit the present discussion to closed
beams of idealized cross-section.
The problem of axial constraint may be conveniently divided into two parts. In the
first, the shear stress distribution due to an arbitrary loading is calculated exclusively
at the built-in end of the beam. In the second, the stress (and/or load) distributions are
calculated along the length of the beam for the separate loading cases of torsion and
shear. Obviously the shear stress systems predicted by each portion of theory must be
compatible at the built-in end.
Argyris and Dunne showed that the calculation of the shear stress distribution at a
built-in end is a relatively simple problem, the solution being obtained for any loading
and beam cross-section by statics. More complex is the determination of the stress
distributions at sections along the beam. These stresses, for the torsion case, are
shown to be the sum of the stresses predicted by elementary theory and stresses
caused by systems of self-equilibrating end loads. For a beam supporting shear
loads the total stresses are again the sum of those corresponding to elementary bend-
ing theory and stresses due to systems of self-equilibrating end loads.
For an n-boom, idealized beam, Argyris and Dunne found that there are n - 3 self-
equilibrating end load, or eigenload, systems required to nullify n - 3 possible modes
of warping displacement. These eigenloads are analogous to, say, the buckling loads
corresponding to the different buckled shapes of an elastic strut. The fact that,
generally, there are a number of warping displacements possible in an idealized
beam invalidates the use of the shear centre or flexural axis as a means of separating
torsion and shear loads. For, associated with each warping displacement is an axis of
twist that is different for each warping mode. In practice, a good approximation is
obtained if the torsion loads are referred to the axis of twist corresponding to the
lowest eigenload. Transverse loads through this axis, the zero warping axis, produce
no warping due to twist, although axial constraint stresses due to shear will still be
present.
In the special case of a doubly symmetrical section the problem of separating the
torsion and bending loads does not arise since it is obvious that the torsion loads
may be referred to the axis of symmetry. Double symmetry has the further effect of
dividing the eigenloads into four separate groups corresponding to (n/4) - 1 pure
flexural modes in each of the xz and yz planes, (44) pure twisting modes about
the centre of symmetry and (44) - 1 pure warping modes which involve neither
flexure nor twisting. Thus, a doubly symmetrical six boom beam supporting a
single shear load has just one eigenload system if the centre boom in the top and
bottom panels is regarded as being divided equally on either side of the axis of
symmetry thereby converting it, in effect, into an eight boom beam.
It will be obvious from the above that, generally, the self-equilibrating stress
systems cannot be proportional to the free warping of the beam unless the free
warping can be nullified by just one eigenload system. This is true only for the four
boom beam which, from the above, has one possible warping displacement. If, in
addition, the beam is doubly symmetrical then its axis of twist will pass through
the centre of symmetry. We note that only in cases of doubly symmetrical beams
do the zero warping and flexural axes coincide.
A further special case arises when the beam possesses the properties of a Neuber
beam (Section 9.5) which does not warp under torsion. The stresses in this case are