The Design of Modern Steel Bridges - TEDI

110 The Design of Modern Steel Bridges
Figure 5.5 Buckling behaviour of thin plates.
5.4.1 Elastic critical buckling of plates
5.4.1.1 Plates under uniaxial compression
Consider an ideally flat residual-stress-free rectangular plate simply supported
along its four edges and subjected to a compressive load F per unit length
uniformly distributed along two opposite edges, as shown in Fig. 5.6. At a
certain value of F the flat form of equilibrium becomes unstable and the plate
buckles; this instability is due to the fact that the energy of the plate in a buckled
form is equal to or less than that if it remained flat under the same edge forces.
The critical value of F may be determined by considering a deflected shape of
the plate consistent with its boundary conditions. One such shape is given by
o ¼ d sin mpx npy
sin ð5:17Þ
a b
where o is the out-of-plane deflection at point (x, y). In this method, the
bending energy U of the plate is equated with the work done T by the applied
forces due to the shortening of the plate. It can be shown[5] that
U ¼ p4 abD
8
d 2 m2 n2
þ
a2 b2 2