56 
When this plastio wave reaches the centre of the panel, the whole plate has 
becoms stretched into the form of a cone, ARA' in section, which is at 
rest and no further deformation or motion will ocour, the whole of the 
original kinetic energy having been converted into permanent plastic 
stretching of the plate. For a rectangular panel of plating given an 
initial uniform velocity, the 
corresponding final deformation 
would be of the roof-top shape 
shown in fig.30, with the 
Sloping sides equally inclined 
to the base; the same shape 
with the top cut off gives the 
intermediate shape whilst de- 
forming, a central flat 
portion being undeformed and 
moving with its original 
velocity whilst the outer 
stretched portion would be at 
rest. 
149. Other special oases 
which can be solved are 
analogous to problems of an 
elastic membrane subjected to 
dynamic loading. Such 
membranes have certain definite Fig. 30 - Shape of deformed 
shapes (theoretically infinite rectangular panel 
in number) in each of which they 
can vibrate with 48 certain definite frequency. Such a vibration is known 
as a normal mode.'’ For the analogous plastic pansl, solutions can be 
obtained if the applied impulsive pressure is distributed over the panel in 
the shape corresponding to any one definite normal mode, provided also that 
variation of the overall magnitude of the pressure with tims is such that ths 
plate continues at all points 
until the final deformed shape is 
obtained by the plate coming to 
rest at all points simltaneously. 
In fig.31, for example, a 
rectangular panel of plating of 
sides 2a and 2b with axes Ox, 
Oy parallel to the sideSand with 
origin at 0 is subjected to a 
suddenly applied pressure 
distribution of po cos 77 x/2a 
cos Ti y/2b. It is show at 
Appendix D that if such a plate 
is fixed around the periphery 
and the deflections are 
relatively small compared with 
the smaller span (one sixth or Pig. 31 - Rectangular plate subjected 
leas) then ulsive pressure 
distribution 
w(t) = w,(t) cos LX cos Az Sag | dog .oog, | (55)! 
w(t) = K/2 (4 — cosec t) eee eee ees (59) 
where w(t) = deflectiono? point (x,y) in the plate at time t 
w,(t) = central deflection of plate at tim t 
h = plate thickness 
6, = uniform membrane stress 
t time 
