Lateral buckling of beams - The Behaviour and Design of Steel Structures to EC3 - page 230

where

π 2 EI z
L 2

GI t + π 2 EI w
L 2

M zx = , (6.3)

and in which EI z is the minor axis flexural rigidity, GI t is the torsional rigidity,
and EI w isthewarpingrigidityofthebeam.Equation6.3showsthattheresistance
to buckling depends on the geometric mean of the flexural stiffness EI z and the
torsional stiffness ( GI t + π 2 EI w / L 2 ). Equations 6.2 and 6.3 apply to all beams
which are bent about an axis of symmetry, including equal flanged channels and
equal angles.
Equation 6.3 ignores the effects of the major axis curvature d 2 w / d x 2 =
− M cr / EI y , and produces conservative estimates of the elastic buckling moment
equalto √ [ ( 1 − EI z / EI y ) { 1 − ( GI t + π 2 EI w / L 2 )/ 2 EI y }] timesthetruevalue.This
correctionfactor,whichisjustlessthanunityformanybeamsectionsbutmaybe
significantly less than unity for column sections, is usually neglected in design.
Nevertheless, its value approaches zero as I z approaches I y so that the true elastic
buckling moment approaches infinity. Thus an I-beam in uniform bending about
itsweakaxisdoesnotbuckle,whichisintuitivelyobvious.Research[1]hasindi-
catedthatinsomeothercasesthecorrectionfactormaybeclosetounity,andthat
it is prudent to ignore the effect of major axis curvature.

6.2.1.2 Beams with unequal end moments

Asimply supported beam with unequal major axis end moments M and β m M is
shown in Figure 6.4a. It is shown in Section 6.12.1.2 that the value of the end

Equation 6.5

3

m M

K = 0.05

M

2

L

K = 3.0

Equation 6.6

(a) Beam

1

m M

M

-1.0 -0.5 0 0.5 1.0

End moment ratio m

(b) Bending moment

(c) Moment factors m

Figure 6.4 Buckling of beams with unequal end moments.