23 - 



507 



and tnence from (lOa) and (ll6) we find: 



u Kj' Kj K 



ug^a 



(119) 



In order to evaluate K we havs now lo assign specific forms to F^, F^ and F . For F^^ 

 the final shape of the defl;ctsd pl-;te in Box Medcl trials suggests as reasonable alternatives: 



F,(r) = - Cos 



' 2 2 



F,(r) = 3 (i_ ,Z) 



(130) 



for which from (117) we find K = 9.3 or 9 respectively. 



Similarly, the final shape of pull-in at the edges in Box Model trials suggests that the 

 same forms are rerisonible shaoes f-r F .Iso, i.e., we m^.y tal<e: 



Fjr) = ^ C.5 

 -* 2 



Tr_r 

 2 



(121) 



or Fj(r) 



-^ (1- r') 

 2 



which give K = 1 . 23 zr 1. 2 respectively. 



Thus the four combinations of (120) and (l2l) give values of K^ K ranging from 10.8 to 

 11, u. The sn-allness of this ran^ie indicates tnat K is not very sensitive to the precise forms for 

 F and F . 



The choice of a form for F is not £o ^csy since we have no direct experimental evidence 

 to help and we therefore seek guidance from the exact solutions given by Taylor(l) for a circular 

 and an elliptical panel onder uniform static pressure. 



We shall thus assume a two-term polynomial: 

 F (r) = p (r - r3) - r^ 



(122) 



which satisfies (113) 2nd (115) while tne p term is similar to that given by Taylor(l) both for the 

 racial displacements in a circular pan=l and the centre-line displacements in an elliptical panel 

 under uniform static pressure. 



In order to ;stirate p we note from (n) and (1I6) that: 



!l_ = i * ^ 



«ab a b 



(123) 



(112) 



Ic, since w is even in x and y the central areal strain A (say) in the plate is from (ill) and 



^{^^ii]..o ii'l]'-^'"''^^' 



(12U) 



y = 



Now from the report "The distortion under pressure of a diaphragm which is clamped along 

 its edge and stressed oeyonj the elastic limit", it can also be shown that the central areal strain 

 is twice the mesn areal strain for Dotn the circular and elliptical panels under uniform static 

 pressure. This suggest? th-at w*; ?.ssuir.e the same r'.-.tio to hold 'in our oroolem for tne final areal 

 strain ut centre ..nd 11= n ^real strain 3u^ to stritcning, i.e., we assume: 



(A,: 



(126) 



whence from (l23) anJ (l2U) we cbtain: 



F'2(0) Fj(0) 



(126) 



