Lateral buckling of beams - The Behaviour and Design of Steel Structures to EC3 - page 238

stresses. The moment at which inelastic buckling initiates depends mainly on
themagnitudeoftheresidualcompressivestressesatthecompressionflangetips,
where yielding causes significant reductions in the effective rigidities ( EI z ) t and
( EI w ) t .Theflange-tipresidualstressesarecomparativelyhighinhot-rolledbeams,
especiallythosewithhighratiosofflangetowebarea,andsotheinelasticbuckling
isinitiatedcomparativelyearlyinthesebeams , asshowninFigure6.2.Theresid-
ualstressesinhot-rolledbeamsdecreaseawayfromtheflangetips(seeFigure3.9
for example), and so the extent of yielding increases and the effective rigidities
steadily decrease as the applied moment increases. Because of this, the inelastic
buckling moment decreases in an approximately linear fashion as the slenderness
increases, as shown in Figure 6.2.
In beams fabricated by welding flange plates to web plates, the compressive
residualstressesattheflangetips,whichincreasewiththeweldingheatinput,are
usually somewhat smaller than those in hot-rolled beams, and so the initiation of
inelastic buckling is delayed, as shown in Figure 6.2. However, the variations of
theresidualstressesacrosstheflangesaremorenearlyuniforminweldedbeams,
andso,onceflangeyieldingisinitiated,itspreadsquicklythroughtheflangewith
little increase in moment. This causes large reductions in the inelastic buckling
moments of stocky beams, as indicated in Figure 6.2.
When a beam has a more general loading than that of equal and opposite end
moments, the in-plane bending moment varies along the beam, and so when
yielding occurs its distribution also varies. Because of this the beam acts as if
non-uniform, and the torsion equilibrium equation becomes more complicated.
Nevertheless, numerical solutions have been obtained for some hot-rolled beams
with a number of different loading arrangements [19, 20], and some of these
(for unequal end moments M and β m M ) are shown in Figure 6.10, together with
approximate solutions given by

0.3 ( 1 − 0.7 M p / M cr )
( 0.61 − 0.3 β m + 0.07 β m )

M I
M p = 0.7 +

(6.23)

in which M cr is given by equations 6.4 and 6.5.
In this equation, the effects of the bending moment distribution are included
in both the elastic buckling resistance M cr = α m M zx through the use of the end
momentratio β m inthemomentmodificationfactor α m ,andalsothroughthedirect
use of β m in equation 6.23. This latter use causes the inelastic buckling moments
M I toapproachtheelasticbucklingmoment M cr astheendmomentratioincreases
towards β m = 1.
Themostseverecaseisthatofequalandoppositeendmoments( β m =− 1),for
whichyieldingisconstantalongthebeamsothattheresistancetolateralbuckling
is reduced everywhere. Less severe cases are those of beams with unequal end
moments M and β m M with β m > 0,whereyieldingisconfinedtoshortregionsnear
the supports, for which the reductions in the section properties are comparatively
unimportant.Theleastseverecaseisthatofequalendmomentsthatbendthebeam