54 
the smaller span, the load-deflection curve for the plate under any given 
distribution of static pressure will then be of tks straight line form 
indicated in fig.27. The linear load-deflection relation more usually 
associated with elastic problems”, thus holds for dishing of a panel of 
plating which is stretohing plastically, It is this result which is the 
essential justification for the linear resistance assumed in the theories 
discussed earlier. 
443. In this discussion of plastic yielding many approximations have been 
introduced and it might fairly be asked - may not the cumlative error be 
so large as to make the final results valueless? The answer to this 
question lies in experiment. Using a large box model with an unsupported 
plate area 6 ft. x4 ft., a steel plate 4 in. thick was tested under al 
uniform statio pressure applied hydraulioally from the inside of the box, 
The results were in good agreement with the soap bubble approximation, the 
shape of the dished plate being similar to that expected of a soap bubble 
and, apart from a small stiffening effect (attributed to som strain 
hardening), the pressure/deflection curve was sensibly linear. Further 
confirmation of the theory was obtained by a static test on a + scale 
replica of the large box model,” 
444. It is important to notice that the soap bubble approximation is only 
valid in problems such as that of the clamped panel of flat plating in which 
the plate is stretching in both directions. For a curved panel of plating 
Such as a submarine pressure hull, in which the plate may be compressing in 
one direction and stretching in the other direotion due to lateral pressure, 
then either equation 51 or 52 leads to an energy absorption approximately 
@qual to So times the numerically greater strain, This approximation has 
been suggested as of possible application to the problem of the submarines 
pressure hull but no serious attempt to do this has yet been made, 
Plastio deformation under dynamic loads 
445. The only problems of plastic yielding under dynamic loads which are 
at all amenable to exact theoretical treatment are simple cases in which ths 
motion is effectively one-dimensional. The one case which has been treated 
fairly exhaustively is that of a long wire fixed at one end and subjeoted to 
a suddenly applied load at the other end, The basic analysis for this case 
was given independently in this country and in Amrica!**’ This case is not 
of direct application to ths 
present problem of beams and 
plates but two general quali- 
tative results are of some 
interest. First, if the stress 
atrain relation under dynamis 
loading is ooncave to the strain | 
axis, for example AKF' in fig, 22, L 
the higher the stress ths more 
slowly is it propagated along 
the wire. Fig,28 illustrates 
a plastic wave travelling along 
the wire tending to become less 
steep as it progresses. The 
stress distribution curve ABC 
becomss the curve A'B'C' as the 
wave travels to the right. 
This is the opposite effect to 
that holding for the shook 
wave in water where the wave 
tends to become more steep as Fig. 28 ~ Plastio wavs in wire 
a D 
m= A linear resistance is not always obtained in elastic problems. 
In particular for the present case of a panel dished by lateral pressure, 
elastio stretohing of the plate introduces a non-linear term in the 
resistance, 
c 
STRESS 
A Al 
STRAIN 
