Beam-columns - The Behaviour and Design of Steel Structures to EC3 - page 315

β m M

1.0 1.0

N

0.8 0.8

0.6 0.6

N

M

N / N cr,z
0.00
0.25
0.50
0.75
0.95

N / N cr,z

0.4 0.4

0.00
0.25
0.50
0.75
0.95

0.2 0.2

(1- )/2 m m

(1- )/2

-1.0 -0.5 0 0.5 1.0 -1.0 -0.5 0 0.5 1.0

End moment ratio m
2 EI w /GI t L 2 = 0.09

End moment ratio m

(a) (b)

2 EI w /GI t L 2 =1.0

Figure 7.16 Flexural-torsional buckling coefficients 1 /α bc in equation 7.45.

7.3.1.3 Beam-columns with elastic end restraints

Theelasticflexural-torsionalbucklingofsymmetricallyrestrainedbeam-columns
with equal and opposite end moments ( β m =− 1) is analysed in Section 7.6.2.
The particular example considered is one for which the minor axis bending and
end warping restraints are equal. This corresponds to the situation in which both
ends of both flanges have equal elastic restraints whose stiffnesses are such that

Flange end moment

Flange end rotation = − EI z

R
1 − R

(7.47)

L

inwhichtherestraintparameter R variesfrom0(flangeunrestrained)to1(flange
rigidly restrained).
It is shown in Section 7.6.2 that the elastic buckling values M cr , MN and N cr , MN
oftheendmomentsandaxialloadforwhichtherestrainedbeam-columnbuckles
elastically are given by

1 − N cr , MN
N cr , z

1 − N cr , MN
N cr , T

M cr , MN
i p N cr , z N cr , T =

(7.48)

in which

N cr , z = π 2 EI z / L cr , (7.49)