of an Elastic Plate under Normal Pressure. 107 



probably much too difficult to be solved completely. The 

 introduction o£ the function cp has spoilt the linearity of 

 Poisson's equations. It will be observed, however, that the 

 equations are linear as far as is concerned. The physical 

 interpretation of this is that, when one set of values of the 

 stresses has been found for a given state of bending of the 

 plate, any set of mean stresses P 1? P 2 , and S, which would 

 be in equilibrium if the plate were not bent, could be super- 

 posed on the set already obtained, and these new stresses 

 would satisfy the differential equations. The pressure p is 

 also altered since the added mean stresses, because they 

 exist in a bent plate, could support a pressure on the surface, 

 just as the tension in a membrane can support a pressure. 



Owing to the complexity of the equations there are very 

 few problems that we can solve directly; that is to say, 

 there are few cases in which we can find w and (/> from a 

 .given value of p. The inverse process of finding p when w 

 is assumed offers, however, no great difficulty. We shall 

 first express our equations in a form that applies to symmetry 

 about the 2-axis, and then make use of these equations to 

 find jo corresponding to a given w. We shall also solve the 

 problem of a rectangular plate bent into a surface of 

 revolution by forces applied at one pair of edges only. The 

 second of the following problems illustrates the importance 

 of the maximum displacement w in its effect on the pressure. 



Symmetry about the z-axis. 



Let x 2 +y 2 = r 2 , and let us suppose that w and <j> are 

 functions of r only. Let P l9 Mj, etc., bear the same relation 

 to the radius as they bore in the earlier equations to the 

 .r-axis. Then, u being the radial displacement, the two 

 longitudinal strains are 



du 1 /dio\ 2 



and the shear strain is zero on an element with sides along 

 and perpendicular to a radius. 



The expressions for the mean stresses in terms of <£ are 



P E !# P E ^i S = . . . (22) 

 r dr dr z 



