58 
fixed round the periphery, the deflection w(t) at time t would be assumed to 
be of the form 
W(t) = W(t) P(xpy) -0- cee nee wee = (63) 
where f(x,y) is any suitable 
shape which vanishes round the 
edge of the plate and is unity 
at the centre so that wo(t) is 
the oentral deflection at tim 
t. For practical purposes, 
the final permanent defleotion 
representing damage is of 
particular importanoe and it 
is natural to choose f(x,y) 
to correspond approximately 
to final observed dishes in 
box model experiments. These 
are in general more pointed 
that the relatively flat 
Soap bubble shape obtained 
in static tests under uniform 
pressure and a reasonable 
approximation to ths observed 
dishes is 
Fig.32 - Empirical shape of a deformed 
rectangular plate 
f(x,y) - - hf I... coe ve (64) 
corresponding to parabolic sections parallel to the sides as shown in fig. 32. 
152.. Using the membrane approximation of a uniform tensions,, and assuming 
that the deflections are reasonably small compared with ths smaller span, 
expressions for the energy p absorbed in plastic deformation by stretching 
from the flat plate to the dished shape, the total energy @ of the deforming 
plate and the work W done by the pressure in deforming the plate are derived 
at Appendix D. In particular, it is shown that for ths parabolic shape of 
equation 6) 
pe Bf e+bbe tm,” jae: qhaue: eeeeee) 
153. The assumptions of proportional motion and a uniform membrane stress 
in the plate lead to a motion® analogous to that of a mass Mon a spring of 
atiffness k subjected to a dynamic load F(t). There is of course the 
important proviso that the plastic deformation is irreversible; subject 
to certain restrictions on the variation of F(t) with time, this can simply 
be taken into account by assuming the motion to cease when wdt) reaches its 
maximm value Wo corresponding to zero velocity of the plate. 
m This is valid in particular, if the pressure distribution remains 
constant after application or if the pressure is transient and osases 
while the plate is still deflecting. 
