Page 170 - Aircraft Stuctures for Engineering Student

154 Structural instability

so that the differential equation of bending of the column is

d2v PcR
-+-v=o
dzz EI
The well-known solution of Eq. (6.2) is
v = Acospz+ Bsinpz
where p2 = PcR/EI and A and B are unknown constants. The boundary conditions
for this particular case are v = 0 at z = 0 and 1. Thus A = 0 and
Bsinpl= 0
For a non-trivial solution @e. v # 0) then
sinpl=O or pl=n.rr wheren= 112131...

giving

or

Note that Eq. (6.3) cannot be solved for v no matter how many of the available
boundary conditions are inserted. This is to be expected since the neutral state of
equilibrium means that v is indeterminate.
The smallest value of buckling load, in other words the smallest value of P which
can maintain the column in a neutral equilibrium state, is obtained by substituting
n = 1 in Eq. (6.4). Hence

Other values of PCR corresponding to n = 2,3,. . . are

These higher values of buckling load cause more complex modes of buckling such as
those shown in Fig. 6.3. The different shapes may be produced by applying external
restraints to a very slender column at the points of contraflexure to prevent lateral
PCR_ -
movement. If no restraints are provided then these forms of buckling are unstable
and have little practical meaning.

1/2
I-
PCH = 4r2EI/L2 PCR= ~T~EI/L'
Fig. 6.3 Buckling loads for different buckling modes of a pin-ended column.