Page 183 - Aircraft Stuctures for Engineering Student

6.5 Energy method 167

plate buckling modes. For the pin-ended column under discussion a deflected form of
m nm
w = xA,sin- (6.49)
n=l I
satisfies the boundary conditions of

and is capable, within the limits for which it is valid and if suitable values for the
constant coefficients A, are chosen, of representing any continuous curve. We are
therefore in a position to find PCR exactly. Substituting Eq. (6.49) into Eq. (6.48) gives
403
EI
u+v=-J (5) (xn’A,sinE) 2 d~
20 n= 1 I

-kj: (;)2( (6.50)
2
n=l
The product terms in both integrals of Eq. (6.50) disappear on integration, leaving
only integrated values of the squared terms. Thus

$PCRTnZA:
4 2
U + V = -En A, - - (6.51)
r4EI 03
413 41
n=1 n=1
Assigning a stationary value to the total potential energy of Eq. (6.51) with respect to
each coefficient A, in turn, then taking An as being typical, we have

from which

2 EIn2
PCR =- as before.
P
We see that each term in Eq. (6.49) represents a particular deflected shape with a
corresponding critical load. Hence the first term represents the deflection of the
column shown in Fig. 6.12, with PCR = 7?EI/I2. The second and third terms
correspond to the shapes shown in Fig. 6.3, having critical loads of 42EI/I2 and
97?EI/I2 and so on. Clearly the column must be constrained to buckle into these
more complex forms. In other words the column is being forced into an unnatural
shape, is consequently stiffer and offers greater resistance to buckling as we observe
from the higher values of critical load. Such buckling modes, as stated in Section
6.1, are unstable and are generally of academic interest only.
If the deflected shape of the column is known it is immaterial which of Eqs (6.47) or
(6.48) is used for the total potential energy. However, when only an approximate
solution is possible Eq. (6.47) is preferable since the integral involving bending
moment depends upon the accuracy of the assumed form of w, whereas the corre-
sponding term in Eq. (6.48) depends upon the accuracy of d2w/dz2. Generally, for
an assumed deflection curve w is obtained much more accurately than d2w/dz2.