Page 189 - Aircraft Stuctures for Engineering Student

6.7 Inelastic buckling of plates 173

Substituting m = 1, we have a/b = fi = 1.414, and for m = 2, a/b = v% = 2.45 and
so on.
For a given value of a/b the critical stress, oCR = Nx,CR/t, is found from Eqs (6.55)
and (5.4). Thus

OCR = (6.57)

In general, the critical stress for a uniform rectangular plate, with various edge sup-
ports and loaded by constant or linearly varying in-plane direct forces (N.y, N,,) or
constant shear forces (N1,) along its edges, is given by Eq. (6.57). The value.of k
remains a function of a/b but depends also upon the type of loading and edge
support. Solutions for such problems have been obtained by solving the appropriate
differential equation or by using the approximate (Rayleigh-Ritz) energy method.
Values of k for a variety of loading and support conditions are shown in Fig. 6.16.
In Fig. 6.16(c), where k becomes the shear buckling coeficient, b is always the smaller
dimension of the plate.
We see from Fig. 6.16 that k is very nearly constant for a/b > 3. This fact is
particularly useful in aircraft structures where longitudinal stiffeners are used to
divide the skin into narrow panels (having small values of b), thereby increasing
the buckling stress of the skin.

For plates having small values of b/t the critical stress may exceed the elastic limit of
the material of the plate. In such a situation, Eq. (6.57) is no longer applicable since,
as we saw in the case of columns, E becomes dependent on stress as does Poisson's
ratio u. These effects are usually included in a plasticity correction factor r] so that
Eq. (6.57) becomes

ffCR = (6.58)
12( 1 - "2)

where E and u are elastic values of Young's modulus and Poisson's ratio. In the
linearly elastic region 11 = 1, which means that Eq. (6.58) may be applied at all
stress levels. The derivation of a general expression for r] is outside the scope of
this book but one2 giving good agreement with experiment is

r]=-- l--u~E,[l -+- l(1 -+-- 3Et)i]
1-u;E 2 2 4 4Es

where Et and E, are the tangent modulus and secant modulus (stress/strain) of the
plate in the inelastic region and ue and up are Poisson's ratio in the elastic and inelastic
ranges.