uu 



131 



and 



W = naa^hz ^ 



These formulas relate the pressure applied, the deflection at the center, and 

 the energy absorbed by a thin plate with constant tension ao^h. 



They may be rewritten in the following form to secure non- 

 dimensionality 



pa ' 4 jj 



1 + 



(1)^ 



[18] 



a TT [19] 



n a ha^ a' 



Equations [l8] and [19] suggest the use of p ' = pa/o^h as the non- 

 dimensional coordinate corresponding to pressure, W' = W/na^h(r^ as the non- 

 dimensional coordinate corresponding to energy, and z/a as the non- 

 dlraenslonal coordinate corresponding to center deflection. We term p' the 

 proportionate pressure and W the proportionate energy. Wis the energy per 

 cubic inch of plate material per unit ultimate stress. 



Although Equations [l8] and [19] were based on membrane formulas It 

 is profitable again to dissociate mentally the diaphragm from a membrane and 

 treat W = W/na^hau and z/a as important intrinsic quantities associated with 

 a thin plate regardless of whether the plate behaves as a membrane or not. 

 If we then plot experimentally determined values of W against z/a, we obtain 

 an accurate and assumptionless curve showing the behavior of a particular 

 plate. The assumption that this curve will be valid for all thin plates is 

 less exacting than the assumption that the plate is a membrane, and therefore 

 more likely to be valid than Equations [18] and [191- 



It Is readily shown, moreover, by similitude considerations that 

 when pa/a^h is plotted against z/a for a thin plate of a given material, the 

 curve for any other thin plate of Identical composition should be the same 

 except for secondary variations caused by bending effects. These will vary 

 with relative thickness and their Influence is less in the plastic range than 

 in the elastic range. Another effect which may be of importance with thin 

 plates of different thicknesses is surface hardening. Thinner plates would 

 be affected to a greater degree by such hardening. 



Of course it Is not enough Just to use non-dimensional quantities 

 to bring the data from different cases into agreement. Thus Young's modulus 

 of elasticity could not be used in place of the plastic quantity, ultimate 

 stress. An elastic quantity, even though of the dimensions of stress, would 

 certainly not serve for the desired unification of the data. 



