Lateral buckling of beams - The Behaviour and Design of Steel Structures to EC3 - page 231

moment M cr atelasticflexural-torsionalbucklingcanbeexpressedintheformof

M cr = α m M zx , (6.4)

in which the moment modification factor α m which accounts for the effect of
the non-uniform distribution of the major axis bending moment can be closely
approximated by

α m = 1.75 + 1.05 β m + 0.3 β m ≤ 2.56, (6.5)

or by

1 /α m = 0.57 − 0.33 β m + 0.10 β m ≥ 0.43.

(6.6)

Theseapproximationsformthebasisofaverysimplemethodofpredictingthe
buckling of the segments of a beam which is loaded only by concentrated loads
applied through transverse members preventing local lateral deflection and twist
rotation.Inthiscase,eachsegmentbetweenloadpointsmaybetreatedasabeam
withunequalendmoments, anditselasticbucklingmomentmaybeestimatedby
using equation 6.4 and either equation 6.5 or 6.6 and by taking L as the segment
length. Each buckling moment so calculated corresponds to a particular buckling
loadparameterforthecompleteloadset,andthelowestoftheseparametersgives
a conservative approximation of the actual buckling load parameter. This simple
methodignoresanybucklinginteractionsbetweenthesegments.Amoreaccurate
method which accounts for these interactions is discussed in Section 6.8.2.

6.2.1.3 Beams with central concentrated loads

Asimply supported beam with a central concentrated load Q acting at a distance
− z Q abovethecentroidalaxisofthebeamisshowninFigure6.5a.Whenthebeam
buckles by deflecting laterally and twisting, the line of action of the load moves
with the central cross-section, but remains vertical, as shown in Figure 6.5c. The
casewhentheloadactsabovethecentroidismoredangerousthanthatofcentroidal
loading because of the additional torque − Qz Q φ L / 2 which increases the twisting
of the beam and decreases its resistance to buckling.
It is shown in Section 6.12.1.3 that the dimensionless buckling load
QL 2 / √ ( EI z GI t ) varies as shown in Figure 6.6 with the beam parameter
K = √ (π 2 EI w / GI t L 2 ) (6.7)

and the dimensionless height ε of the point of application of the load given by

EI z
GI t

ε = z Q
L

. (6.8)