Lateral buckling of beams - The Behaviour and Design of Steel Structures to EC3 - page 260

analysedforitsin-planemomentdistribution(themomentmodificationfactorsof
equation 6.13 or Figure 6.7 may be used) and for an effective length L cr equal to
thesegmentlength L .Theso-determinedelasticbucklingmomentofeachsegment
is then used to evaluate a corresponding beam load set, and the lowest of these
is taken as the elastic buckling load set. This method produces a lower bound
estimate which is sometimes remarkably close to the true buckling load set.
However, this is not always the case, and so a much more accurate but still
reasonablysimplemethodhasbeendeveloped[36].Inthismethod,theaccuracyof
thelowerboundestimate(obtainedasdescribedabove)isimprovedbyallowingfor
theinteractionsbetweenthecriticalsegmentandtheadjacentsegmentsatbuckling.
This is done by using a simple approximation for the destabilising effects of the
in-plane bending moments on the stiffnesses of the adjacent segments, and by
approximatingtherestrainingeffectsofthesesegmentsonthecriticalsegmentby
usingtheeffectivelengthchartofFigure3.21aforbracedcompressionmembersto
estimatetheeffectivelengthofthecriticalbeamsegment.Astep-by-stepsummary
[36] is as follows:

(1) Determine the properties EI z , GI t , EI w , L of each segment.
(2) Analyse the in-plane bending moment distribution through the beam, and
determine the moment modification factors α m for each segment from
equation 6.13 or Figure 6.7.
(3) Assume all effective length factors k cr are equal to unity.
(4) Calculate the maximum moment M cr in each segment at elastic buckling
from

π 2 EI z
L cr

GI t + π 2 EI w
L cr

M cr = α m (6.67)

with L cr = L , and the corresponding beam buckling loads Q s .
(5) Determine a lower-bound estimate of the beam buckling load as the lowest
value Q ms of the loads Q s , and identify the segment associated with this as
the critical segment 12. (This is the approximate method [45] described in
the preceding paragraph.)
(6) Ifamoreaccurateestimateofthebeambucklingloadisrequired,usetheval-
ues Q ms and Q rs 1 , Q rs 2 calculatedinstep5togetherwithFigure6.25(which
is similar to Figure 3.19 for braced compression members) to approximate
the stiffnesses α r 1 , α r 2 of the segments adjacent to the critical segment 12.
(7) Calculate the stiffness of the critical segment 12 from 2 EI zm / L m .
(8) Calculate the stiffness ratios k 1 , k 2 from

2 EI zm / L m
0.5 α r 1, r 2 + 2 EI zm / L m .

k 1,2 = (6.68)

(9) Determine the effective length factor k cr for the critical segment 12 from
Figure 3.21a, and the effective length L cr = k cr L .