Local buckling of thin-plate elements - The Behaviour and Design of Steel Structures to EC3 - page 106

Simply supported Simply supported

b

cr

cr

Free edge

L

Figure 4.8 Buckled pattern of a plate free along one edge.

4.2.1.2 Plates free along one longitudinal edge

The thin flat plate shown in Figure 4.8 is simply supported along both transverse
edges and one longitudinal edge, and is free along the other. The differen-
tial equation of equilibrium of the plate in a buckled position is the same
as equation 4.105 (see Section 4.8.1). The buckled shape which satisfies this
equation differs, however, from the approximately square buckles of the simply
supported plate shown in Figure 4.7.The different boundary conditions along the
free edge cause the plate to buckle with a single half wave along its length, as
shown in Figure 4.8. Despite this, the solution for the elastic buckling stress σ cr
can still be expressed in the general form of equation 4.3 in which the buckling
coefficient k σ is now approximated by

b
L

2

k σ = 0.425 + , (4.9)

as shown in Figure 4.9.
Forthelong-plateelementswhichareusedasflangeoutstandsinmanystructural
steelmembers,thebucklingcoefficient k σ isclosetotheminimumvalueof0.425.
In this case the elastic buckling stress (for a steel for which E = 210000 N/mm 2
and ν = 0.3) is equal to the yield stress f y when

b
t

f y
235 = 18.5.

(4.10)

Once again it is more economical to use longitudinal stiffeners to increase the
elastic buckling stress than transverse stiffeners. The stiffness requirements of
intermediate and edge longitudinal stiffeners are discussed in Section 4.2.1.1.

4.2.1.3 Plates with other support conditions

The edges of flat-plate elements may be fixed or elastically restrained, instead
of being simply supported or free. The elastic buckling loads of flat plates with