495 



since this merely states that we =re me->surinj w in the direction of applied load. 



We also bear in mind that the constants A^ B^ C^, A^ B^ C^, A^ as defined earlier by equations 

 (9), (13) and (26) are all positive. 



In order to consiaer which types of ni:tion are possibla, we first collect together some 

 consequences of our physical assumptions. 



Firstly, the assumption that the plate stretches under constant tension T^ implies that: 



T = T when S^ > 



; (32) 



T 4 T when S ^ 

 



and similarly for the edge pull-in we have: 



T' = T, when S, > 



* ' (33) 



T- -^ T^ when S^ ^ 

 Secondly, our assumption that the edge fixing is relatively the weaker implies that: 

 T^ > T^ > (31) 



and thence as a corollary from (32), (33). (3t) and (27) we have: 



S > when S^ > (35) 



Finally, we shall need to show that there is no tendency of the plate or edges to buckle cr compress 

 since this is not physically obvious. For this purpose we can assume our general equations (2l) and 

 (27) to hold provided T and T" are compressive and not tensile, i.e., we must have: 



T 4 when S^ < 1 



(3«) 



T' ■?: when S, < J 



since otherwise we should be extracting energy from the plate or edge fixings which Is contrary to cur 

 assumption that the dishing is plastic with Irreversible absorption of energy. 



A. 6 Continuity condi tions . 



On physical grounds the displacements U, V, W and therefore S, S , S must be continuous; We 

 must also assume that T and T' are finite and then from (?7) it follows that S Is always finite and 

 therefore that S^ is continuous. Also, if w J- 0, then from (30) and (31) S will be finite or positively 

 Infinite, the latter corresponding to the concept of an Instantaneous impulse; thus S will be either 

 continuous or will Increase discont inuously provided W ^ 0. 



The continuity conditions are thus: 



W, S , S , S continuous. 



w, S continuous cr increasing discont inuously. 



(37) 



It follows that if at the end of any particular type of motion we have: 

 W ^ 



(38) 



So >' 



Sj i 

 then these same relations will be satisfied by the initial conditions for the succeeding type of motion. 



A.7 



