Page 186 - Aircraft Stuctures for Engineering Student

170 Structural instability

area of the plate. For the analysis we may conveniently employ the method of total
potential energy since we have already, in Chapter 5, derived expressions for strain
and potential energy corresponding to various load and support configurations. In
these expressions we assumed that the displacement of the plate comprises bending
deflections only and that these are small in comparison with the thickness of the
plate. These restrictions therefore apply in the subsequent theory.
First we consider the relatively simple case of the thin plate of Fig. 6.14, loaded
as shown, but simply supported along all four edges. We have seen in Chapter 5
that its true deflected shape may be represented by the infinite double trigonometrical
series
mnx nry
w= 2 TA,sin- a Sinb
m=l n=l
Also, the total potential energy of the plate is, from Eqs (5.37) and (5.45)

The integration of Eq. (6.52) on substituting for w is similar to those integrations
carried out in Chapter 5. Thus, by comparison with Eq. (5.47)

The total potential energy of the plate has a stationary value in the neutral equili-
brium of its buckled state (Le. N, = Nx,CR). Therefore, differentiating Eq. (6.53)
with respect to each unknown coefficient A, we have

and for a non-trivial solution
1 m2 n2 '
Nx,CR = 220- -+- (6.54)
m2 ( a2 b2)
Exactly the same result may have been deduced from Eq. (ii) of Example 5.2, where
the displacement w would become infinite for a negative (compressive) value of N,
equal to that of Eq. (6.54).
We observe from Eq. (6.54) that each term in the infinite series for displacement
corresponds, as in the case of a column, to a different value of critical load (note,
the problem is an eigenvalue problem). The lowest value of critical load evolves
from some critical combination of integers m and n, i.e. the number of half-waves
in the x and y directions, and the plate dimensions. Clearly n = 1 gives a minimum
value so that no matter what the values of m, a and b the plate buckles into a half