Page 464 - Aircraft Stuctures for Engineering Student

11.2 Built-in end of a closed section beam 445

the elementary torsion theory stresses since no constraint effects are present. When
bending loads predominate, however, it is generally impossible to design an efficient
structure which does not warp.
In this chapter the calculation of spanwise stress distributions in closed section
beams is limited to simple cases of beams having doubly symmetrical cross-sections.
It should be noted that simplifications of this type can be misleading in that some of
the essential characteristics of beam analysis, for example the existence of the n - 3
self-equilibrating end load systems, vanish.

Bear stress distribution at a built-in end of a
xed section beam

This special case of structural constraint is of interest due to the fact that the shear
stress distribution at the built-in end of a closed section beam is statically determinate.
Figure 11.1 represents the cross-section of a thin-walled closed section beam at its
built-in end. It is immaterial for this analysis whether or not the section is idealized
since the expression for shear flow in Eq. (9.39), on which the solution is based, is
applicable to either case. The beam supports shear loads S,x and Sy which generally
will produce torsion in addition to shear. We again assume that the cross-section
of the beam remains undistorted by the applied loads so that the displacement of
the beam cross-section is completely defined by the displacements u, v, w and the rota-
tion 8 referred to an arbitrary system of axes Oxy. The shear flow q at any section of
the beam is then given by Eq. (9.40), that is

d0 du dv
- + - cos$ + - sin$ + -
dz dz dz

Fig. 11.1 Cross-section of a thin-walled beam at the built-in end.