Page 138 - Aircraft Stuctures for Engineering Student

Bending of thin plates

Generally, we define a thin plate as a sheet of material whose thickness is small
compared with its other dimensions but which is capable of resisting bending, in
addition to membrane forces. Such a plate forms a basic part of an aircraft structure,
being, for example, the area of stressed skin bounded by adjacent stringers and ribs in
a wing structure or by adjacent stringers and frames in a fuselage.
In this chapter we shall investigate the effect of a variety of loading and support
conditions on the small deflection of rectangular plates. Two approaches are
presented: an ‘exact’ theory based on the solution of a differential equation and an
energy method relying on the principle of the stationary value of the total potential
energy of the plate and its applied loading. The latter theory will subsequently be
used in Chapter 6 to determine buckling loads for unstiffened and stiffened panels.

The thin rectangular plate of Fig. 5.1 is subjected to pure bending moments of
intensity M, and My per unit length uniformly distributed along its edges. The
former bending moment is applied along the edges parallel to the y axis, the latter
along the edges parallel to the x axis. We shall assume that these bending moments
are positive when they produce compression at the upper surface of the plate and
tension at the lower.
If we further assume that the displacement of the plate in a direction parallel to the
z axis is small compared with its thickness t and that sections which are plane before
bending remain plane after bending, then, as in the case of simple beam theory, the
middle plane of the plate does not deform during the bending and is therefore a
neutralplane. We take the neutral plane as the reference plane for our system of axes.
Let us consider an element of the plate of side SxSy and having a depth equal to the
thickness t of the plate as shown in Fig. 5.2(a). Suppose that the radii of curvature of
the neutral plane n are px and pv in the xz and yz planes respectively (Fig. 5.2(b)).
Positive curvature of the plate corresponds to the positive bending moments which
produce displacements in the positive direction of the z or downward axis. Again,
as in simple beam theory, the direct strains E, and E), corresponding to direct stresses
a, and oy of an elemental lamina of thickness Sz a distance z below the neutral plane