Lateral buckling of beams - The Behaviour and Design of Steel Structures to EC3 - page 237

compressionmember.Thevariationofthedimensionlesslimitingmoment M L / M y
for η = λ 2 / 4 is shown in Figure 6.9, in which λ = √ ( M y / M zx ) plotted along the
horizontalaxisisequivalenttothegeneralisedslendernessratiousedinFigure3.4
for an elastic compression member. Figure 6.9 shows that the limiting moments
of short beams approach the yield moment M y , while for long beams the limiting
moments approach the elastic buckling moment M zx .

6.3 Inelastic beams
The solution for the buckling moment M zx of a perfectly straight simply sup-
ported I-beam with equal end moments given by equations 6.2 and 6.3 is only
valid while the beam remains elastic. In a short-span beam, yielding occurs
before the ultimate moment is reached, and significant portions of the beams
are inelastic when buckling commences.The effective rigidities of these inelastic
portions are reduced by yielding, and consequently, the buckling moment is also
reduced.
For beams with equal and opposite end moments ( β m =− 1), the distribution
of yield across the section does not vary along the beam, and when there are no
residualstresses,theinelasticbucklingmomentcanbecalculatedfromamodified
form of equation 6.3 as

π 2 ( EI z ) t
L 2

( GI t ) t + π 2 ( EI w ) t

M I = (6.22)

L 2

in which the subscripted quantities ( ) t are the reduced inelastic rigidities which
are effective at buckling. Estimates of these rigidities can be obtained by using
the tangent moduli of elasticity (see Section 3.3.1) which are appropriate to the
varying stress levels throughout the section. Thus the values of E and G are used
in the elastic areas, while the strain-hardening moduli E st and G st are used in
the yielded and strain-hardened areas (see Section 3.3.4). When the effective
rigidities calculated in this way are used in equation 6.22, a lower bound esti-
mate of the buckling moment is determined (Section 3.3.3). The variation of the
dimensionless buckling moment M / M y with the geometrical slenderness ratio
L / i z of a typical rolled-steel section which has been stress-relieved is shown in
Figure 6.2. In the inelastic range, the buckling moment increases almost linearly
with decreasing slenderness from the first yield moment M y = W el , y f y to the
full plastic moment M p = W pl , y f y , which is reached soon after the flanges are
fullyyielded,beyondwhichbucklingiscontrolledbythestrain-hardeningmoduli
E st , G st .
Theinelasticbucklingmomentofabeamwithresidualstressescanbeobtained
inasimilarmanner,exceptthatthepatternofyieldingisnotsymmetricalaboutthe
sectionmajoraxis,sothatamodifiedformofequation6.76foramonosymmetric
I-beam must be used instead of equation 6.22. The inelastic buckling moment
varies markedly with both the magnitude and the distribution of the residual