Lateral buckling of beams - The Behaviour and Design of Steel Structures to EC3 - page 273

or

M L = M y − δ 0 N cr , z
M zx

1 + d f
2

W el , y
W el , z

N cr , z
M zx

M L
1 − M L / M zx

in which N cr , z = π 2 EI z / L 2 , M L = f L W el , y is the limiting moment at first yield,
and M y = f y W el , y is the nominal first yield moment. This can be solved for the
dimensionlesslimitingmoment M L / M y .Inthecasewherethecentralcrookedness
δ 0 is given by

δ 0 N cr , z
M zx

W el , z / W el , y
1 + ( d f / 2 )( N cr , z / M zx )

= θ 0 = η , (6.21)

in which η defines the magnitudes δ 0, θ 0 , then the dimensionless limiting moment
simplifies to

M L
M y =

1

(6.18)

Φ 2 − λ 2

Φ +

in which

Φ = ( 1 + η + λ 2 )/ 2, (6.19)

and

λ = ( M y / M zx ) (6.20)

is a generalised slenderness.

6.13 Appendix - effective lengths of beams

6.13.1 Beams with elastic end restraints

ThebeamshowninFigure6.18isrestrainedatitsendsagainstminoraxisrotations
d v / d x and against warping rotations ( d f / 2 ) d φ/ d x , and the boundary conditions
at the end x = L / 2 can be expressed in the form of

( d v / d x ) L / 2 = − EI z

M B + M T 2 R 2

1 − R 2 ,

L

and

( d f / 2 )( d φ/ d x ) L / 2 = − EI z

M T − M B 2 R 4

1 − R 4 , L