Lateral buckling of beams - The Behaviour and Design of Steel Structures to EC3 - page 270

and the torsion equation is

d x − EI w d 3 φ

GI t d φ

d x 3 = M y d v

d x − V z v .
These equations reduce to equations 6.81 and 6.82 for the case of equal and
opposite end moments ( β m =− 1).
Closedformsolutionsoftheseequationsarenotavailable,butnumericalmeth-
ods[3-11]havebeenused.Thenumericalsolutionscanbeconvenientlyexpressed
in the form of

M cr = α m M zx (6.4)

in which the factor α m accounts for the effect of the non-uniform distribution of
the bending moment M y on elastic lateral buckling. The variation of α m with the
end moment ratio β m is shown in Figure 6.4c for the two extreme values of the
beam parameter

π 2 EI w
GJL 2

K = (6.7)

of 0.05 and 3.Also shown in Figure 6.4c is the approximation

α m = 1.75 + 1.05 β m + 0.3 β m ≤ 2.56.

(6.5)

6.12.1.3 Beams with central concentrated loads

The major axis bending moment M y and the shear V z in the beam with a central
concentrated load Q shown in Figure 6.5 are given by

M y = Qx / 2 − Q x − L / 2
V z = Q / 2 − Q x − L / 2 0 ,

in which the values of the second terms are taken as zero when the values inside
the Macaulay brackets are negative. When the beam buckles, the minor axis
bending equation is

EI z d 2 v

d x 2 =− M y φ ,
and the torsion equation is

d x − EI w d 3 φ

GI t d φ

d x 3 = Q 2 ( v − z Q φ) L / 2 ( 1 − 2 x − L / 2 0 ) + M y d v d x − V z v

inwhich z Q isthedistanceofthepointofapplicationoftheloadbelowthecentroid,
and ( Q / 2 )( v − z Q φ) L / 2 is the end torque.