Beam-columns - The Behaviour and Design of Steel Structures to EC3 - page 313

1.0

0.8

0.2

0.5

0.6

1.0

0.4

2.0

0.2

N cr,z / N cr,T = 5.0

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
M cr,MN / M zx

Figure 7.14 Elastic buckling load combinations for beam-columns with equal end moments.

sothat ( 1 − N cr , MN / N cr , T )>( 1 − N cr , MN / N cr , z )( 1 − N cr , MN / N cr , y ) .Inthiscase,
equation 7.39 can be safely approximated by the interaction equation

N cr , MN
N cr , z

1
( 1 − N cr , MN / N cr , y )

M cr , MN
M zx

+ = 1. (7.40)

7.3.1.2 Beam-columns with unequal end moments

The elastic flexural-torsional buckling of simply supported beam-columns with
unequal major axis end moments M and β m M has been investigated numerically,
and many solutions are available [7-11].The conservative interaction equation

2

M / √ F
M E

N
N cr , z

+ = 1 (7.41)

has also been proposed [9], in which

M E = π 2 EI z GI t / L 2 . (7.42)
Thefactor1/ √ F inequation7.41varieswiththeendmomentratio β m asshownin
Figure7.15,andallowstheunequalendmomentstobetreatedasequivalentequal
end moments M / √ F . Thus the elastic buckling of beam-columns with unequal
end moments can also be approximated by modifying equation 7.35 to
( M cr , MN / √ F ) 2
M zx

1 − N cr , MN
N cr , T

1 − N cr , MN
N cr , z

= , (7.43)

or by a similar modification of equation 7.40.