12 
where Ww, = K is the permanent central deflection (that is of 0) and w is 
the permanent deflection of any point (x, y). Equation D7, therefore, 
represents permanent deformation and the plate will remain in equilibrium 
thereafter since the applied pressure p, cos'b* cosSf is insufficient to 
cause further stretching. Thus, equation D7 with K defined by equation D5 
is the_final en dishing produced by the suddenly applied pressure 
TX 
p, cos Ba 00S SE 
Energy absorbed in plastic deforaation 
D2. # The energy{) p absorbed in plastic deformation by stretching from the 
flat plate to the dished shape is 8,h times the increase in area of the 
Plate. If proportional motion of the deforming plate be assumed, that is 
the plate deforms in a constant shape throughout the motion, then the 
deflection wt) for any point (x, y) in the plate at time t would be of the 
form 
WARE EWPOCIEELCYS Iv) L) sae ‘elec. see) “eae, woe) Qemeubies (DS) 
where wt) is the central deflection at time t 
therefore Oe. 2B h wel t) | | (of)? + go" | dydx eee) (D9) 
—a- 
In particular, if the plate is assumed to deform in the empirical parabolic 
shape given by 
f(x, y) = fiz} {1-4 ose “eet cee Pet ae a CDIO) 
then Ope & {e+e Bhiwe(t): coset k., Woven aes OSD) 
Kinetic energy of deforming plate 
D3. The total kinetic energy CYof the deforming plate at tine t is 
B= snp | [[Mon] oe dual t))? SP aed Ck ei es) 
For the shape given by equation Di0 this gives 
OF = pn an ey’ eae" ccceMitieea  Bocee css CDIS) 
