Plates of Variable Thickness. 959 



Case II. — 2 he Complete Plate under Axial Pressure. 



As a second example consider a thin circular plate, nearly 

 plane, clamped at the outer edge and subject to an axial 

 force P. Here a small force P yields a relatively large 

 displacement, and our method is put to a more delicate test. 

 Again adopt cylindrical coordinates (fig. 3). 



Fig. 3. 



The plate is not restricted to be symmetrical. Accordingly 

 let a and ft be proportional to the distances of the bases of 

 the plate from the plane 0=0, so that z = ta and z = t/3 are 

 the equations of the lower and upper bases respectively. 



The boundary conditions are the following : — 



fU(0, 0) = 0, ti<0,0) =e, 



w(a, v) = 



Br 



0. 



fJ>+ 2 "[ ; 





dr. 



Again, we take the formula for potential energy as tli 

 point of departure. The term in t' 1 is now 



-dz. 

 This term can only be made to vanish by setting 



w = w (r), U = U (r), 



and the boundary conditions are not thereby violated. 



The constant term in W disappears and we can write the 

 integrand of the term in t as the sum of squares : — 



+ 



2/*[U ' 2 



+ 



IV 



+ 



\bz) 1 



+ A* 





In the present case we are led to attempt to make this inte- 

 grand vanish, since the case of an ordinary plane plate shows 



