120 Dr. J. jprescott on tlie Equations of Equilibrium 



means of its bending. That is to say, we should be assuming 

 that the flexural rigidity is negligible and that the plate 

 behaves as a membrane with a non-uniform tension. We 

 are justified in neglecting the terms containing the factor h 

 when iv is everywhere small in comparison with h, and we 

 are justified in neglecting the terms containing li z when h is 

 small compared with the maximum value of w, unless the 

 bent middle surface nearly coincides with a developable 

 surface, in which case h must be small in comparison with 

 the maximum deflexion measured from this developable 

 surface. 



We are now going to assume that a bent plate is 

 symmetrical about the 2-axis and that the terms containing 

 h z in (14) are negligible. With these assumptions, equations 

 (12) and (14) become 



Cl ( d .} CIW d 2 W . Q x 



^f^ v *>} = -*. 5^' • • ; (79) 



d 



d ( dw ' d<f> 



di 



I ( dw ' d<j> ) p 



r\dr ' dr] ~~ ~ l/*Ef * * ' 



One integration of each of these equations gives 



(§0) 



ri^-\&)- ■ ■ ■ (81) 



and ^. ** = -J>-* (82) 



dr dr 4/ifcj 



The constant of integration is zero in each case because 



-,— is zero where r is zero for a circular plate with no 

 dr L 



central hole. 



Now putting 



. dd> , 1 dw 



Z = r llr' 6= rlh' S = ? "' 



equations (81) and (82) become 



5— s' (83) 



and '*=~&- • • .■ • • ( ^ 



When 6 is eliminated from these the equation for f is 



cP l - JL J-- s ! 

 ds<~ 128/t 2 E a f 



