486 - 2 - 



By analogy with the plate assumption, we assjme that the edge fixings exert a mean tension 

 T per unit length of edge when pull-in is occurring, and we assume 



corresponding to the edge fixings being relatively weaker than the plate. 



We neglect any elastic effects, all energy absorption in plate or at edges thus being 

 assumed Irreversible. 



The plate is assumed to be Initially plane and at rest. 

 The impulsive pressure Is assumed to be of the form 

 P = P (t) f;, (x,y) 



Where x,y are rectangular co-ordinates In the plane of the plate and t is time, and the solution Is 

 obtained for the two specific cases of P(t): 



(a) Load conmunicated instantaneously as an initial velocity. 



(b) Load suddenly applied and then maintained constant, i.e., P(t) a> H(t) where H(t) 

 is Heaviside's unit function. 



(2) Method of analysis . 



Using the preceding assumptions an approximate theory is developed in the Appendix, using 

 the method of an earlier report, the Jeflectlon w and the Jlsplacements, u, v In the plane of the 

 plate being assumed to be of constant distribution In space as given by equations (l) to (3) In the 

 Appendix. 



The deflection w and displacements u, v are assumed sufficiently small to neglect terms 

 higher than second order in w or first order In u and v. 



As a further simplification, the solutions are carried through on the assumption that 

 (T - T )/T is small, this being justified later by application of the theory to the experimental 

 results. 



(3) Nature of motion and main results of theory . 



For the particular loads assumed. It is shown in the Appendix that the motion will consist of 

 successive stages of one or other of the following types :- 



Type I: Plate stretching and pulling-ln at edges. 



Type III Plate Inextensional but pulllng-in at edges. 



The interesting qualitative result is then obtained that for either type of load the motion 

 will consist entirely of Type II if the load Is of sufficiently small magnitude corresponding to 

 sufficiently small final mean deflections w . This result Is exhibited In Figure 2, where it Is 

 seen that below a certain critical final mean deflection W or W (= 1.89 W ) depending on type of 

 load, all the increase of area S^ occurs at the edges implylr^ that all the energy Is absorbed there. 

 Above the critical value W or W , a progressively smaller fraction of S occurs at the edges and 

 consequently more and more energy Is absorbed in plate stretching. 



It may be noted that the two curves In Figure 2 for the two different types of load are for 

 practical purposes of the same shape over the range plotted , since. If the curve (b) were plotted with 

 abscissa *^/W . it becomes virtually coincident with curve ^). This is a somewhat remarkable fact, 

 since the equations for the two curves are not mathematically Identical and no obvious physical reason 

 can be suggested for such near-coincidence. However, this result does suggest that the qualitative 

 shape of the curve Is not very dependent on the particular form of loading assumed In the analysis. 



In 



