Page 177 - Aircraft Stuctures for Engineering Student

6.3 Effect of initial imperfections 161

where X2 = P/EI. The final deflected shape, v, of the column depends upon the form
of its unloaded shape, vo. Assuming that
X nrz
v0 = A, sin- 1 (6.24)
n=l
and substituting in Eq. (6.23) we have

-+ d2v X-v -- $gn2A,sin- n7rz
dz2 1
n= 1
The general solution of this equation is
v = Bcos Xz + D sin Xz + ca n2An sin nm
- I
n=l n2 - a
where B and D are constants of integration and cy = X212/2. The boundary condi-
tions are v = 0 at z = 0 and I, giving B = D = 0 whence
3o n’~, nrz
u = =sinI (6.25)
R=l
Note that in contrast to the perfect column we are able to obtain a non-trivial solution
for deflection. This is to be expected since the column is in stable equilibrium in its
bent position at all values of P.
An alternative form for a is

(see Eq. (6.5))
Thus a is always less than one and approaches unity when P approaches PCR so that
the first term in Eq. (6.25) usually dominates the series. A good approximation,
therefore, for deflection when the axial load is in the region of the critical load is
(6.26)

or at the centre of the column where z = 1/2

(6.27)

in which AI is seen to be the initial central deflection. If central deflections
6(= v - AI) are measured from the initially bowed position of the column then
from Eq. (6.27) we obtain

which gives on rearranging

(6.28)
and we see that a graph of 6 plotted against 6/P has a slope, in the region of the critical
load, equal to PCR and an intercept equal to the initial central deflection. This is the