Lateral buckling of beams - The Behaviour and Design of Steel Structures to EC3 - page 239

Elastic buckling M cr

1.1

m = 1.0 0.5 0 0.5 1.0 - - Full plasticity M p

1.0

= 1.0 0.5 0 0.5 1.0

-

m -

0.9

0.8

M

m

M

L

0.7

Computer solution (254 UB 31)
Approximate equation 6.23

0.6

0.5

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Generalised slenderness ( M p / M cr )

Figure 6.10 Inelastic buckling of beams with unequal end moments.

in double curvature ( β m = 1), for which the moment gradient is steepest and the
regions of yielding are most limited.
The range of modified slenderness √ ( M p / M cr ) for which a beam can reach
the full plastic moment M p depends very much on the loading arrangement. An
approximate expression for the limit of this range for beams with end moments
M and β m M can be obtained from equation 6.23 as
M p
M cr

0.39 + 0.30 β m − 0.07 β m
0.70

= . (6.24)

p

In the case of a simply supported beam with an unbraced central concen-
trated load, yielding is confined to a small central portion of the beam, so
that any reductions in the section properties are limited to this region. Inelas-
tic buckling can be approximated by using equation 6.23 with β m =− 0.7 and
α m = 1.35.

6.4 Real beams

Real beams differ from the ideal beams analysed in Section 6.2.1 in much the
same way as do real compression members (see Section 3.4.1). Thus any small
imperfections such as initial crookedness, twist, eccentricity of load, or horizon-
tal load components cause the beam to behave as if it had an equivalent initial
crookedness and twist (see Section 6.2.2), as shown by curve A in Figure 6.11.
Ontheotherhand,imperfectionssuchasresidualstressesorvariationsinmaterial