Page 462 - Aircraft Stuctures for Engineering Student

Structural constraint

The analysis presented in Chapters 9 and 10 relies on elementary theory for the deter-
mination of stresses and displacements produced by axial loads, shear forces and
bending moments and torsion. Thus, no allowance is made for the effects of restrained
warping produced by structural or loading discontinuities in the torsion of open or
closed section beams, or for the effects of shear strains on the calculation of direct
and shear stresses in beams subjected to bending and shear.
In this chapter we shall examine some relatively simple examples of the above
effects; more complex cases require analysis by computer-based techniques such as
the finite element method.

Structural constraint stresses in either closed or open beams result from a restriction
on the freedom of any section of the beam to assume its normal displaced shape
under load. Such a restriction arises when one end of the beam is built-in although
the same effect may be produced practically, in a variety of ways. Thus, the root
section of a beam subjected to torsion is completely restrained from warping into
the displaced shape indicated by Eq. (9.52) and a longitudinal stress system is induced
which, in a special case discussed later, is proportional to the free warping of the
beam.
A slightly different situation arises when the beam supports shear loads. The stress
system predicted by elementary bending theory relies on the basic assumption of
plane sections remaining plane after bending. However, for a box beam comprising
thin skins and booms, the shear strains in the skins are of sufficient magnitude to
cause a measurable redistribution of direct load in the booms and hence previously
plane sections warp. We shall discuss the phenomenon of load redistribution resulting
from shear, known as shear lag, in detail later in the chapter. The prevention of this
warping by some form of axial constraint modifies the stress system still further.
The most comprehensive analysis yet published of multi- and single cell beams
under arbitrary loading and support conditions is that by Argyris and Dunne'.
Their work concentrates in the main on beams of idealized cross-section and while
the theory they present is in advance of that required here, it is beneficial to examine