Page 182 - Aircraft Stuctures for Engineering Student

166 Structural instability

Yt
Fig. 6.12 Shortening of a column due to buckling.

column. In particular, the energy method is extremely useful when the deflected form
of the buckled column is unknown and has to be ‘guessed’.
First, we shall consider the pin-ended column shown in its buckled position in
Fig. 6.12. The internal or strain energy U of the column is assumed to be produced
by bending action alone and is given by the well known expression

(6.44)

or alternatively, since EZ d2v/& = -M

(6.45)

The potential energy V of the buckling load PCR, referred to the straight position of
the column as the datum, is then
v = -PCR6
where 6 is the axial movement of PcR caused by the bending of the column from its
initially straight position. By reference to Fig. 5.15(b) and Eq. (5.41) we see that

giving

(6.46)

The total potential energy of the column in the neutral equilibrium of its buckled state
is therefore

(6.47)
or, using the alternative form of U from Eq. (6.45)

(6.48)

We have seen in Chapter 5 that exact solutions of plate bending problems are
obtainable by energy methods when the deflected shape of the plate is known. An
identical situation exists in the determination of critical loads for column and thin