52 
to cause plastic yielding and, since elastic defects are being neglected, the 
beam remains undeflected. When W = 4G)/1 however, the moment is Gp at the 
centre and plastic yielding commences at the centre. Away from the centre 
the moment is still less than Gp so that the beam remains unbent and tends, 
therefore, to deflect in the shape shown in fig. 25a, consisting of two 
LOAD 
W=4G,/1 
DEFLECTION 
(a) (b) 
central load 
Fig. 25 - Plastic bending of simply supported beam by concentrated 
straight line portions with a concentrated bend at the centre. Since the 
resisting moment Gp is independent of the amount of bending it follows 
that the deflection will take place under the constant load W = 4G)/1. 
The load deflection curve is thus of the flat-topped shape shown in fig, 25b, 
being similar to that assumed initially ror the stress-strain relation in 
simple tension (or compression) illustrated in fig. 23. Experiments on 
the plastic bending of beams of various sections have shown that this 
simple theory is substantially correct. 
Dishing of a thin panel of plating 
439. Consider a thin circular plate, initially flat, rigidly fixed round 
its circumference and subjected to uniform lateral pressure. For 
appreciable plastic dishing, the bending resistanoe oan be neglected in 
comparison with the resistanoe to stretching since the latter involves 
far more energy absorption for a thin plate. Secondly, the lateral 
pressure will in general be small compared with the yield stresses in the 
plane of the plate so that the problem becomes approximately one of two- 
dimensional stress with one principal stress, normal to the plane of the 
plate, equal to zero. By virtue of symmetry the prinoipal stresses will 
then be 
6, = ©, normal to plate 
6, » radial in plane of plate 
6t , circumferential in plane of plate. 
On these assumptions, with use of the previous plasticity equations and 
neglecting elasticity the problem has been solved, The most important 
result which follows whether equation 51 or equation 52 is taken as the 
condition of plastic yielding, is that 
6-= Ot = Bo eee eee eee eve eee (56) 
This result implies that the stress in the plane of the plate is an equal 
tension in all directions, that is, it behaves like a soap bubble. 
