Page 474 - Aircraft Stuctures for Engineering Student

11.4 Shear lag 455

A problem closely related to the restrained torsion of rectangular section beams is
that generally known as shear fag. We have seen in Chapter 9 that torsion induces
shear stresses in the walls of beams and these cause shear strains which produce warp-
ing of the cross-section. When this warping is restrained, direct stresses are set up
which modify the shear stresses. In a similar manner the shear strains in the thin
walls of beams subjected to shear loads cause cross-sections to distort or warp so
that the basic assumption of elementary bending theory of plane sections remaining
plane is no longer valid. The direct and shear stress distributions predicted by elemen-
tary theory therefore become significantly inaccurate. Further modifications arise
when any form of structural constraint prevents the free displacement of the cross-
sections of a beam. Generally, shear lag becomes a problem in wide, relatively
shallow, thin-walled beams such as wings in which the shear distortion of the thin
upper and lower surface skins causes redistribution of stress in the stringers and
spar caps while the thicker and shallower spar webs experience little effect.
Consider the box beam shown in Fig. 11.9. Elementary bending theory predicts
that the direct stress at any section AA would be uniform across the width of the
covers so that the stringers and web flanges would all be subjected to the same
stress. However, the shear strains at the section cause the distortion shown so that
the intermediate stringers carry lower stresses than the web flanges. Since the resultant
of the direct stresses must be equivalent to the applied bending moment this means
that the direct stresses in the web flanges must be greater than those predicted by
elementary bending theory. Our investigation of the shear lag problem will be
restricted to idealized six- and eight-boom doubly symmetrical rectangular section
beams subjected to shear loads acting in the plane of symmetry and in which the
axis of twist, the flexural axis and the zero warping axis coincide; the shear loads
therefore produce no twist and hence no warping due to twist. In the analysis we
shall assume that the cross-sections of beams remain undistorted in their own plane.
Figure 11.10 shows an idealized six-boom beam built-in at one end and carrying a
shear load at the other; the corner booms have a cross-sectional area B while the
central booms have a cross-sectional area A. At any section the vertical shear load

Fig. 11.9 Shear distortion in the covers of a box beam.