Lateral buckling of beams - The Behaviour and Design of Steel Structures to EC3 - page 246

anadditionalrigidrestraintatitscentrewhichpreventslateraldeflectionandtwist
rotation, then its buckled shape is given by
φ
(φ) L / 4 = ( v ) L / 4 = sin π x

v

L / 2 ,

(6.33)

and its elastic buckling moment is given by

π 2 EI z
( L / 2 ) 2

GI t + π 2 EI w
( L / 2 ) 2

M cr = . (6.34)

Theendsupportsofabeammayalsodifferfromsimplesupports.Forexample,
bothendsofthebeammayberigidlybuilt-inagainstlateralrotationabouttheminor
axis and against end warping. If the beam has equal and opposite end moments,
then its buckled shape is given by

φ
(φ) L / 2 = ( v ) L / 2 = 1

v

1 − cos π x
L / 2

, (6.35)

2

and its elastic buckling moment is given by equation 6.34.
In general, the elastic buckling moment M cr of a restrained beam with equal
and opposite end moments ( β m =− 1.0) can be expressed as

π 2 EI z
L cr

GI t + π 2 EI w

M cr = , (6.36)

L cr

in which

L cr = k cr L (6.37)

is the effective length and k cr is an effective length factor.
When a beam has several rigid restraints which prevent local lateral deflection
and twist rotation, then the beam is divided into a series of segments. The elastic
bucklingofeachsegmentmaybeapproximatedbyusingitslengthastheeffective
length L cr .Onesegmentwillbethemostcritical,andtheelasticbucklingmoment
of this segment will provide a conservative estimate of the elastic buckling resis-
tance of the whole beam. This method ignores the interactions between adjacent
segmentswhichincreasetheelasticbucklingresistanceofthebeam.Approximate
methods of allowing for these interactions are given in Section 6.8.
The use of the effective length concept can be extended to beams with loading
conditions other than equal and opposite end moments. In general, the maximum
moment M cr at elastic buckling depends on the beam section, its loading, and its
restraints, so that

π 2 EI w

GI t L 2 , loading, 2 z Q

M cr L
√ ( EI z GI t ) = fn

d f , restraints

. (6.38)

A partial separation of the effect of the loading from that of the restraint
conditions may be achieved for beams with centroidal loading ( z Q = 0) by