The Design of Modern Steel Bridges - TEDI

The strain energy of the elastic restraint at midpoint ¼ sa 2 =2, where s is the
stiffness of the restraint.
Work done by the axial force ¼ p 2 Pa 2 =4L, wherefrom
and
Pcr ¼ p2 EIc
L 2
1 þ 2sL3
p 4 EIc
leff ¼ L 1 þ 2sL3
p 4 EIc
5.4 Local buckling of plate elements
Rolled Beam and Plate Girder Design 109
, but not more than 4p 2 EIc=L 2
1=2
, but not less than 1
2 L:
If a beam is made up of thin plate elements, i.e. thin web or flanges, then these
plate elements may buckle well before the beam section reaches its overall
elastic or buckling strength. Elastic buckling theories may be applied to derive
the critical buckling stress of individual plate elements in the beam crosssection,
i.e. the magnitude of the applied stress at which an ideal initially flat
residual-stress-free plate becomes unstable and deflects out of its initially flat
plane. The critical buckling stress depends upon the pattern of the applied
stress, the geometry of the plate and the out-of-plane restraints on its edges.
However, unlike overall buckling of beams and columns, a slender plate
element may carry increased loading beyond the elastic critical value with
increased out-of-plane deflection, i.e. it may have post-buckling strength.
A plate with some initial out-of-flatness starts deflecting out-of-plane right
from the beginning of load application, and the rate of deflection increases as
the critical buckling stress is reached; in the post-buckling range the stiffness
of the plate is less than that below the critical buckling level. The stiffness and
strength of a plate element in the post-buckling range depend on the in-plane
restraint at the edges of the plate. As a plate element starts deflecting out-ofplane,
the distribution of in-plane stresses due to applied load becomes nonuniform
and, in addition, bending stresses develop. As the combined in-plane
and bending stresses reach the elastic limit in some parts of the plate, these
parts lose their stiffness. The ultimate strength of the plate element is reached
when a large part of the plate has yielded. Residual stresses in parts of the plate
due to welding or rolling may bring about an earlier onset of yielding in these
parts and may lower both the ultimate strength of the plate and its stiffness in
the post-elastic range.