of an Elastic Plate under Normal Pressure. Ill 



by equation (37) is applied about each unit length of the 

 rim, and that a variable pressure p given by equation (35) 

 is applied to that surface of the plate which becomes the 

 convex surface in the bent state. 



Fig. 5 shows the way the pressure acts across a diametral- 

 section, and shows also the direction of the bending moment 

 on the rim. 



Fig-. 5. 



"r-^ 



rt ^n l i i i i r> 



Ml, 



The usual Poisson or Kirchhoff method would make the 

 pressure p and the tensions Y 1 and P 2 zero, but would give 

 the same values of M x and M 2 . 



In the solution we have just obtained we assumed that P 2 

 was zero at the edge of the plate, and this gave b 2 = 2a 2 . If, 

 however, Pi is not zero, but has a positive value at the rim, 

 we have only to leave b 2 in our equations, and for this case 

 b 2 will be greater than 2a 2 . In this way a constant term 

 will be added to each of the quantities P 1? P 2 , and p. The 

 added quantities are just the stresses and pressure in a 

 stretched membrane which has no flexural rigidity. 



Problem 2. — To find the pressure required, on the present 

 theory, to produce the deflexion which a circular disk 

 .assumes according to Poisson's theory when subjected to a 

 uniform pressure and supported without clamping at its 



edge. 



According to Love ( 4 Theory of Elasticity/ 3rd edition, 

 Art. 314) or Morley (' Strength of Materials/ Art. 149), the 

 deflexion of a disk of radius a due to a uniform pressure p l is 



iv=-H(2b 2 r 2 -r±) + Q, .... (39) 



- h - *=m (± ^ • (*» 



and b 2 =^±^a\ (41) 



and C is a constant which depends on the position of 

 the origin. If the xy plane touches the middle surface 

 at the centre, then the constant G is zero. We shall find 

 the pressure on the present theory which will produce the 

 deflexion given in (39). 



