Lateral buckling of beams - The Behaviour and Design of Steel Structures to EC3 - page 235

in which the central initial crookedness δ 0 and twist rotation θ 0 are related by

δ 0
θ 0 =

M zx
π 2 EI z / L 2 ,

(6.15)

then the deformations of the beam are given by

v

δ = φ θ = sin π x L ,

(6.16)

in which

δ 0 = θ

δ M / M zx

1 − M / M zx ,

θ 0 =

(6.17)

asshowninSection6.12.2.Thevariationsofthedimensionlesscentraldeflection
δ/δ 0 and twist rotation θ/θ 0 are shown in Figure 6.8, and it can be seen that
deformationbeginsatthecommencementofloading,andincreasesrapidlyasthe
elastic buckling moment M zx is approached.
The simple load-deformation relationships of equations 6.16 and 6.17 are of
the same forms as those of equations 3.8 and 3.9 for compression members
with sinusoidal initial crookedness. It follows that the Southwell plot tech-
nique for extrapolating the elastic buckling loads of compression members from
experimental measurements (see Section 3.2.2) may also be used for beams.

1.2

1.0

0.8

Straight beam 0 = f 0 = 0
equations 6.1 and 6.3

0.6

Beam with initial crookedness and twist
0 / 0 = f 0 / 0 = sin x / L
0 / f 0 = M zx / N cr,z

equation 6.17

0.4

0.2

0

0 5 10 15 20 25

Dimensionless deformation / 0 or / 0

Figure 6.8 Lateral deflection and twist of a beam with equal end moments.