of an Elastic Plate under Normal Pressure. 113 



the centre by p , we get 



Wq — Ha 2 ( 2b 2 — a 2 ) 



= Ha 4 (2s-l), (46) 



where 5= — = - (47) 



a* l + <r v 



Also p = Pl + SBBEhb 2 



=Pi+|h 3 EW(6s 3 -4s 2 -M) 



3—;"'° "(2,-1)2 



1 



J 

 If o" = 3-, this becomes 



1 2 ?r 2 6V-4^2-M) 



A Jl+_(l_a)^-— ^-j, . (48) 



/,o=Pi{l + l-07g|)"|, .... (49) 



which shows that p is approximately twice as great as j) 1 if 

 the maximum deflexion w is equal to the thickness of the 

 plate. 



Moreover, at the rim of the plate, where r = a, we find 

 that 



p =Pl { 1-0-484 g) 2 } ; . . . (50) 



so that, if w = < 2h, the pressure at the edge of the plate is 

 little more than half of p x . 



The preceding numerical results show how inaccurate are 

 the usual Poisson equations when the maximum deflexion of 

 the plate is of the same order as the thickness. As long as 

 the maximum value of w is less than one-tenth of the 

 thickness, it is clear that Poisson's equations give results that 

 are as accurate as any that we can ever hope to get in such 

 calculations. But for a body like a piece of tin bent so ihat 

 the maximum deflexion is three or four times the thickness 

 of the sheet, Poisson's method does not even give a pressure 

 of the right order. In the example we have just worked 

 out, if w were only four times as great as 2h, the pressure 

 would vary from nearly — 1 p Y at the rim to 17 pi at the 

 centre, whereas Poisson's equations give a uniform pressure 

 p x over the disk. 



Problem 3. — A rectangular plate is bent into a part of a 

 surface of revolution, which is nearly nearly cylindrical, by 

 forces applied at one pair of opposite edges, the other pair 



Phil Mag. S. 6. Vol. 43. No. 253. Jan. 1922. I 



