Compression members - The Behaviour and Design of Steel Structures to EC3 - page 52

N N

L

v

x

N N

(a) Unloaded member (b) Undeflected loaded

member

(c) Deflected member

Figure 3.3 Straight compression member.

The unloaded position of the member is shown in Figure 3.3a. A concentric
axial load N is applied to the upper end of the member, which remains straight
(Figure 3.3b). The member is then deflected laterally by a small amount v as
shown in Figure 3.3c, and is held in this position. If the original straight position
of Figure 3.3b is one of stable equilibrium, the member will return to it when
releasedfromthedeflectedposition,whileiftheoriginalpositionisoneofunstable
equilibrium, the member will collapse away from it when released. When the
equilibrium of the original position is neither stable nor unstable, the member is
inaconditiondescribedasoneofneutralequilibrium.Forthiscase,thedeflected
position is one of equilibrium, and the member will remain in this position when
released.Thus,whentheload N reachestheelasticbucklingvalue N cr atwhichthe
originalstraightpositionofthememberisoneofneutralequilibrium,themember
may deflect laterally without any change in the load, as shown in Figure 3.2b.
The load N cr at which a straight compression member buckles laterally can be
determinedbyfindingadeflectedpositionwhichisoneofequilibrium.Itisshown
in Section 3.8.1 that this position is given by

v = δ sin π x / L (3.1)

in which δ is the undetermined magnitude of the central deflection, and that the
elastic buckling load is

N cr = π 2 EI / L 2

(3.2)

in which EI is the flexural rigidity of the compression member.