The Design of Modern Steel Bridges - TEDI

and stresses caused by lateral deflection and twist combine to cause yielding.
This is the non-linear or ‘divergence’ theory of buckling. The critical bending
moment of the ideal straight beam with very high yield stress will be discussed
first, and then it will be described how this value is modified to take account of
the onset of yielding.
5.3.1 Buckling of an ideal beam
The critical bending moment of a perfectly straight elastic beam with crosssection
symmetrical about both axes is given by
Mcr ¼ p
rffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
EIy GJ
ð5:1Þ
a
where
Le
1 þ p2 EIw
L 2 e GJ
EIy ¼ flexural rigidity about the minor axis
GJ ¼ torsional rigidity
EI w ¼ warping rigidity
Le ¼ half-wavelength of buckling, or ‘effective length’, as it is generally
called
a ¼ is a correction factor, just less than 1.0, to correct for deflection due to
bending; it is given approximately by ðIx IyÞ=Ix, where Ix is the
major axis moment of inertia.
For the standard case of a beam of length L subjected to equal and opposite
end moments, restrained at its ends against lateral deflections and twist but free
to rotate in plan, and without any intermediate lateral restraint, Le is equal to L.
Equation (5.1) can also be expressed as
Mcr ¼ p2 E
L 2 e
Rolled Beam and Plate Girder Design 97
rffiffiffiffiffiffiffiffi
IyIw
a
b ð5:2Þ
where b represents the contribution of the torsional rigidity of the section and
is given by
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b ¼ 1 þ L2e GJ
p2 s
ð5:3Þ
EIw
For equal flange I-sections
h
Iw ¼ Iy
2
4
where h is the distance between the centroids of the flanges; hence equation
(5.2) may be expressed as
Mcr ¼ p2 EIy
2L 2 e
h b pffiffiffi ð5:4Þ
a